Special Session 49: Nonlinear Waves in Discrete Systems

Nonlinearity and Nonreciprocity: Edge States, Wave Dynamics, and Solitons

Vassos Achilleos CNRS-LAUM France Co-Author(s):

Abstract: We present results on the interplay between nonlinearity, nonreciprocity, and topology in non-Hermitian lattices. We demonstrate that nonreciprocal topological lattices support insensitive edge states whose spectral properties remain immune to symmetry-breaking nonlinear perturbations, due to a competition between nonlinear self-interactions and mode nonorthogonality. We then analyze bulk wave-packet dynamics in the nonlinear Hatano-Nelson model, identifying three successive regimes driven by nonreciprocal amplification, and derive analytical predictions showing that nonlinearity modifies both the magnitude and time dependence of wave-packet acceleration. Finally, we show that nonlinearity enables unidirectional soliton formation in active nonreciprocal lattices, demonstrated analytically and experimentally in an active electrical transmission line.

Spectral Precursors of Rogue-Wave Cascades across the Salerno Deformation

Jimmie Adriazola Arizona State University USA Co-Author(s):

Abstract: We study higher-order rogue waves and multi-generation cascades on the Salerno lattice. The model interpolates between integrable Ablowitz-Ladik and DNLS, and we generate cascades from Thomas-Fermi quench data. In the integrable periodic case, rogue waves come from spectral degeneracies at band endpoints on modulationally unstable arcs. We ask whether similar degeneracies continue to organize cascade formation once integrability is broken. The work has two parts. On the lattice, we compute Floquet and Lax spectral portraits of TF data along the Salerno deformation, looking for branch crossings and arc reconnections that precede focusing events. In the continuum limit, we use the Madelung transform and Whitham modulation to track dispersionless focusing, gradient catastrophe, and the onset of oscillations. The goal is a single picture connecting continuum modulation to lattice spectral geometry.

Enabling inverse design of mechanical nonlinear waves

Nicholas Boechler University of California San Diego USA Co-Author(s):

Abstract: In this talk, I will review our recent progress on enabling the ability to answer, within a mechanical context: "For this dynamical application, what is the best nonlinear elastic constitutive response I could choose?" and, given that, "How can I achieve that nonlinearity?" Using an 1D, FPUT-like chain, we conduct an inverse design algorithm to find polynomial nonlinear springs that optimize, separately, the minimization of peak kinetic energy experienced in response to impact, and a prescribed displacement path executed, both evaluated at the opposite end of the chain. We then show how a second inverse design algorithm can be used to identify solid elastic springs that leverage geometric nonlinearity to achieve the targeted spring law. Simulation and experiment results are compared. Future extensions to irreversible contexts, higher dimensions, and mechanical neural networks will be discussed.

Can Breathers Survive? Instability and Control in Periodic Optical Media

Martina Chirilus-Bruckner Leiden University Netherlands Co-Author(s):

Abstract: Motivated by the quest to develop optical storage, we investigate the formation and persistence of breathers in materials with periodically varying properties in the presence of randomness, noise, and damping. A particular focus is on whether appropriately chosen forcing can extend the lifetime of these long-lived structures. Our work builds on the rigorous construction of breathers for a Klein-Gordon equation with periodically varying coefficients and is further motivated by recent results suggesting that such breathers are typically unstable. In this presentation, we explore the extent to which randomness, noise, damping, and forcing can mitigate, delay, or even counteract this instability. Since their stability is notoriously difficult to address analytically, numerical stability analysis is essential. At the same time, the resulting numerical schemes naturally lead to lattice equations whose relationship to the underlying continuum description is rather delicate.

Breather solutions to the coupled Ablowitz-Ladik lattice equations

Baofeng Feng University of Texas Rio Grande Valley USA Co-Author(s):

Abstract: We present our recent results on general breather solutions to the coupled Ablowitz-Ladik (cAL) lattice equations in terms of pfaffians. We first bilinearize the cAL lattice equations into a set of three bilinear equations under nonzero plane wave background. We show that the first two bilinear equations can be derived from the discrete BKP equation through Miwa transformation, while the third one can only be approved via the identities of pfaffians. In the second part, we will present dynamical behaviors of one-, two-breather solutions. It is interesting that a type of resonant breather solutions exists in the cAL lattice equations which seems new in the literature.

Discrete Nonlinear Schr\odinger versus Ablowitz-Ladik: Existence and dynamics of NLS-type lattices over a nonzero background

Nikos I Karachalios Department of Mathematics, University of Thessaly Greece Co-Author(s):    Dirk Hennig, Dionyssios Mantzavinos and Dimitrios Mitsotakis

Abstract: We will discuss the local well-posedness for Ablowitz-Ladik (AL) and Discrete Nonlinear Schr\{o}dinger (DNLS) equations supplemented with a broad class of nonzero boundary conditions and, in addition, derive analytical upper bounds for the minimal guaranteed lifespan of their solutions. These bounds suggest finite-time collapse (blow-up) of solutions in the case of the AL systems and the phenomenon of quasi-collapse in the case of DNLS systems. Numerical simulations confirm another theoretical result on the proximity of the dynamics between the two models over time scales up to the common solution lifespan. Finally, for power nonlinearities, we prove the asymptotic equivalence between the two discrete models in the continuous limit.

Wavepacket Modulation in Nonlinear Lattices: Nonlinear Schrodinger Formalism and Application in Dusty Plasma Crystals

Ioannis Kourakis Khalifa University of Science and Technology United Arab Emirates Co-Author(s):    Ioannis Kourakis, Aysha Nihidha Pulakkal, Mehnaz Gafoor, Hadi Susanto, Nick Lazarides

Abstract: This study focuses on the modulational dynamics of wave packets propagating in a one-dimensional (1D) hybrid Fermi-Pasta-Ulam-Tsingou / Klein-Gordon (FPUT-KG) type lattice chain, incorporating an arbitrary polynomial coupling potential anharmonicity combined with the presence of a nonlinear on-site (substrate) potential. Applying Newell`s multiple scales method, we have derived a Nonlinear Schrodinger type equation (NLSE) and thus obtained analytical expressions for the dispersion and nonlinearity coefficients, in terms of the carrier wavenumber k and the intrinsic lattice configuration parameters. We have explored the conditions for different regimes to occur, in order to determine the type of envelope soliton solutions to be sustained. The analysis is extended to different types of coupling anharmonicity and onsite potential nonlinearity, offering insight into the interplay between these factors and the system's dynamical behavior. We have focused in particular on extreme amplitude envelope modes (freak waves), e.g. of the Peregrine soliton type, and on their dependence on the various intrinsic system parameter values [1]. Our results are relevant in various contexts where periodic systems (e.g. crystals) may occur. As a novel field of application, this research applies to the modeling of dust-lattice waves in dusty plasma crystals. The substrate potential in this case is provided by electrostatic trapping in laboratory experiments, in combination with gravity, while the inter-site interaction potential is essentially of Debye-Hueckel type. This relation with be briefly discussed, and some preliminary results will be presented [2]. [1] Aysha Nihidha Pulakkal et al, in preparation (2026). [2] I. Kourakis & P. K. Shukla, Int. J. Bifurcation & Chaos 16, 1711 (2006).

Exponential asymptotics for calculating stability in discrete NLS equations

Christopher Lustri The University of Sydney Australia Co-Author(s):    Panos Kevrekidis, Jon Chapman, Ines Aniceto

Abstract: Exponential asymptotics can be used to calculate exponentially small asymptotic contributions in singularly-perturbed problems. I will apply these techniques to determine the behaviour of breathers in the discrete nonlinear Schr\odinger equation. I will begin by demonstrating how these ideas predict the well-known feature of the existence of two types of fixed points, namely site-centered and inter-site-centered. I will then show that the exponentially small contributions to the solution can be used to calculate the asymptotic scaling and precise value of the exponentially small eigenvalues in the system associated with site-centered (stable) and inter-site-centered (unstable) configurations. I will then explain how this method can be extended to study related problems, such as the dynamics of kink solutions and solutions with long-range dependence. This method paves the way for such an analysis in a wide range of lattice nonlinear dynamical equation models.

Geometry Meets Computation: Localized States in DNLS Lattices with Extended Interactions

VASSILIOS M ROTHOS Aristostle University of Thessaloniki Greece Co-Author(s):

Abstract: \begin{document} \begin{abstract} Localized wave phenomena in discrete nonlinear Schrodinger (DNLS) lattices play a central role in applications ranging from nonlinear optics to energy transport in complex media. In such systems, the interplay between nonlinearity, discreteness, and extended interactions gives rise to rich families of spatially localized states, commonly referred to as discrete solitons. In this talk, we revisit the existence of stationary localized waves in DNLS lattices with non-nearest neighbour interactions from a dynamical systems perspective. Building on recent results, where such structures are constructed via invariant manifold techniques and homoclinic dynamics, we highlight the geometric mechanisms underlying their formation and organization. We then introduce a complementary computational approach aimed at promoting high-accuracy numerical approximations to mathematically reliable solutions. Without entering into technical details, we outline how modern validation techniques can be used to rigorously certify the existence of localized states while providing quantitative control of approximation errors. The combined geometric and computational viewpoint offers a flexible framework for the analysis of nonlinear lattice models and opens the way to the systematic validation of localized wave phenomena in discrete media. \end{abstract} \end{document}